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Given a problem $X$, to show it is is $\sf NP$-complete, one usually shows a deterministic reduction from an $\sf NP$-complete problem.

If it is hard to show deterministic reduction, then one shows a randomized reduction.

  1. What does a deterministic reduction give about the hardness of the problem that a randomized reduction does not give?

  2. What is the consequence if there is a randomized reduction from an $\sf NP$-complete to $X$, however one can show there is no deterministic reduction from an $\sf NP$-complete problem to $X$?

  3. If we have only randomized reduction, is $X$ still an $\sf NP$-complete problem or could it have faster algorithms?

  4. What does it mean for problem $X$ if there is a randomized reduction from $\sf NP$-complete problems? (Does it mean $X$ is still $\sf NP$-complete?)

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Let's answer your questions one by one:

  1. A deterministic reduction proves NP-hardness. A randomized reduction doesn't. If a problem is NP-hard with respect to randomized reductions, then it could (potentially) be solvable in polynomial time even if P$\neq$NP. The assumption BPP$\neq$NP does rule out polynomial time algorithms for any such problem, however – even randomized ones.

  2. The consequence is that P$\neq$BPP, which is conjectured to be false.

  3. NP-hardness is defined with respect to deterministic reductions.

  4. We say that the problem is "NP-hard with respect to randomized reductions".

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    $\begingroup$ Right. That was my interpretation. $\endgroup$ – Yuval Filmus Jan 1 '15 at 11:22
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    $\begingroup$ You can ask the moderators there to migrate the question. But generally speaking, perhaps it would be a good idea to restrain yourself on asking questions, trying to answer them on your own for longer before asking them online. $\endgroup$ – Yuval Filmus Jan 1 '15 at 11:32
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    $\begingroup$ This site cannot replace basic education on the topic. A good start would be reading some lecture notes on topics that interest you, though it's better to take formal courses with graded assignments. $\endgroup$ – Yuval Filmus Jan 1 '15 at 11:35
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    $\begingroup$ Your question is probably too wide. But this discussion is becoming too long, and it's time to stop it. $\endgroup$ – Yuval Filmus Jan 1 '15 at 11:44
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    $\begingroup$ Not necessarily: the problem could be harder than NP. $\endgroup$ – Yuval Filmus Jan 8 '15 at 12:34

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